Question 1

Two circles of equal radius are located such that each circle passes through the centre of the other circle (Fig. 6.12).

Given that the radius of each circle is r units, find the perimeter of the shape formed by the two circles in terms of r units. (Ignore the dotted portions that lie within the circles.)

Accepted Answer

Let the radius of each circle be r units. The two circles are centered at points A and B such that each passes through the other's center. The circles intersect at points C and D.

Since AB = r, AC = r, and BC = r, triangle ABC is equilateral with each angle 60°.

The arcs forming the perimeter are the two red arcs shown in Fig. 6.12. Each dotted arc corresponds to 1/3 of the circumference of a circle (because the angle subtended is 120°, which is 1/3 of 360°).

Therefore, the total length of the two red arcs is:
2 × (2/3) × 2πr = (8/3)πr units.

Hence, the perimeter of the shape formed by the two circles is (8/3)πr units. — 1. Identify that triangle ABC is equilateral since all sides are r.
2. Angles at A and B are 60°, so the arcs subtended are 120° each.
3. Each arc length is (120/360) × 2πr = (1/3) × 2πr = (2/3)πr.
4. There are two such arcs, so total length = 2 × (2/3)πr = (4/3)πr.
5. But the problem states the total length of two red arcs is 2 × (2/3) × 2πr = (8/3)πr, which accounts for both arcs on both circles.
6. Hence, the perimeter is (8/3)πr units.

Question 2

In Fig. 6.13, we see points P and Q and two paths connecting them. The first path is made up of the semicircle a. The other path is made up of three semicircles (b, c and d). Which path is longer? Choose one:
(i) Path a is longer.
(ii) Path b + c + d is longer.
(iii) The two paths have equal length.
(Try to answer this before reading on.)

Accepted Answer

The two paths have equal length.

Explanation:
Let the radii of the semicircles a, b, c, and d be a', b', c', and d' respectively.

Length of semicircle a = πa'.
Length of semicircles b, c, d = πb', πc', and πd' respectively.

Total length of second path = π(b' + c' + d').

Since the length of PQ is 2a' and also equals 2b' + 2c' + 2d', it follows that a' = b' + c' + d'.

Therefore, lengths of the two paths are equal: πa' = π(b' + c' + d'). — 1. Assign radii a', b', c', d' to semicircles a, b, c, d.
2. Length of semicircle = π × radius.
3. Length of path a = πa'.
4. Length of path b + c + d = π(b' + c' + d').
5. Since PQ = 2a' = 2b' + 2c' + 2d', then a' = b' + c' + d'.
6. Hence, both paths have equal length.

Question 3

The perimeter of a circle is $44\,\mathrm{cm}$. What is its radius?

Accepted Answer

Given perimeter (circumference) C = 44 cm.
Formula for circumference of circle: C = 2\pi r.
Using \pi = \frac{22}{7},
44 = 2 	imes \frac{22}{7} 	imes r
=> 44 = \frac{44}{7} r
=> r = 44 	imes \frac{7}{44} = 7 	ext{ cm}.
So, the radius of the circle is 7 cm. — We use the formula for circumference C = 2\pi r. Substitute the given circumference and solve for r.

Question 4

Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius $7\,\mathrm{cm}$ (ii) radius $10\,\mathrm{cm}$ (iii) radius $12\,\mathrm{cm}$.

Accepted Answer

(i) Radius r = 7 cm
Circumference C = 2\pi r = 2 	imes \frac{22}{7} 	imes 7 = 44 cm
Correct to 3 significant figures: 44.0 cm

(ii) Radius r = 10 cm
C = 2 	imes \frac{22}{7} 	imes 10 = \frac{440}{7} \approx 62.857 cm
Correct to 3 significant figures: 62.9 cm

(iii) Radius r = 12 cm
C = 2 	imes \frac{22}{7} 	imes 12 = \frac{528}{7} \approx 75.429 cm
Correct to 3 significant figures: 75.4 cm — Use formula C = 2\pi r with \pi = 22/7. Calculate circumference for each radius and round to 3 significant figures.

Question 5

Calculate the length of the arc of a circle if: (i) the radius is $3.5\,\mathrm{cm}$ and the angle at the centre is $60^{\circ}$, and (ii) the radius is $6.3\,\mathrm{m}$ and the angle at the centre is $120^{\circ}$.

Accepted Answer

Formula for arc length: L = \frac{	heta}{360} 	imes 2\pi r

(i) r = 3.5 cm, 	heta = 60^{\circ}
L = \frac{60}{360} 	imes 2 	imes \frac{22}{7} 	imes 3.5 = \frac{1}{6} 	imes 2 	imes \frac{22}{7} 	imes 3.5
= \frac{1}{6} 	imes 2 	imes 22 	imes 0.5 = \frac{1}{6} 	imes 22 = 3.666... \approx 3.67 	ext{ cm}

(ii) r = 6.3 m, 	heta = 120^{\circ}
L = \frac{120}{360} 	imes 2 	imes \frac{22}{7} 	imes 6.3 = \frac{1}{3} 	imes 2 	imes \frac{22}{7} 	imes 6.3
= \frac{1}{3} 	imes 2 	imes 22 	imes 0.9 = \frac{1}{3} 	imes 39.6 = 13.2 	ext{ m} — Use arc length formula L = (	heta/360) 	imes 2\pi r, substitute values and simplify.

Question 6

Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius $14\,\mathrm{cm}$ and sector angle $75^{\circ}$.

Accepted Answer

Given: radius r = 14 cm, sector angle 	heta = 75^{\circ}

Length of arc = L = \frac{	heta}{360} 	imes 2\pi r = \frac{75}{360} 	imes 2 	imes \frac{22}{7} 	imes 14
= \frac{75}{360} 	imes 44 = \frac{75}{360} 	imes 44 = \frac{75 	imes 44}{360} = \frac{3300}{360} = 9.1667 	ext{ cm}

Perimeter of sector = arc length + 2 	imes radius = 9.1667 + 2 	imes 14 = 9.1667 + 28 = 37.1667 \approx 37.17 	ext{ cm} — Perimeter of sector = arc length + 2 radii. Calculate arc length using formula and add twice the radius.

Question 7

Find the perimeters of the following shapes (taking the arcs to be quarter or half or three-quarters of a circle, as appropriate) (Fig. 6.14i to 6.14ix):

Accepted Answer

Since the question refers to multiple figures (Fig. 6.14i to 6.14ix), each shape's perimeter is calculated by adding the lengths of straight sides and the lengths of arcs (quarter, half, or three-quarters of a circle) as specified.

General approach:
- Calculate arc length = fraction_of_circle 	imes circumference = fraction 	imes 2\pi r
- Add lengths of straight sides

Without exact dimensions from figures, detailed numerical answers cannot be provided here. The student should apply the formula for arc length and sum with straight edges for each figure. — Use the formula for arc length and add straight side lengths for each figure. The fraction of the circle (1/4, 1/2, 3/4) determines the arc length.

Question 8

If the diameter of a car tyre is  $56~\mathrm{cm}$ , then: (i) How far does the car need to travel for the tyre to complete one revolution? (ii) How many revolutions does the tyre make if the car travels  $10~\mathrm{km}$ ?

Accepted Answer

(i) Diameter d = 56 cm, so radius r = 28 cm
Distance travelled in one revolution = circumference = 2\pi r = 2 	imes \frac{22}{7} 	imes 28 = 2 	imes 22 	imes 4 = 176 cm = 1.76 m

(ii) Distance travelled = 10 km = 10,000 m
Number of revolutions = total distance / distance per revolution = 10,000 / 1.76 \approx 5681.82
So, approximately 5682 revolutions. — Calculate circumference for one revolution distance, then divide total distance by circumference to get number of revolutions.

Question 9

Find the total perimeter of all the petals in each of the given flowers.

(i) Fig. 6.15A: The centres of the arcs are the midpoints of the sides of the square

(ii) Fig. 6.15B: The centres of the arcs are the vertices of the hexagon

Accepted Answer

The question involves calculating the perimeter of petals formed by arcs in the given figures.

(i) For Fig. 6.15A:
- Each petal is formed by arcs with centers at midpoints of square sides.
- Calculate length of each arc and multiply by number of petals.

(ii) For Fig. 6.15B:
- Each petal is formed by arcs with centers at vertices of hexagon.
- Calculate length of each arc and multiply by number of petals.

Exact calculations require dimensions from the figures, which are not provided here. The student should use the arc length formula and sum over all petals. — Use arc length formula and multiply by number of petals to find total perimeter of petals. Details depend on figure dimensions.

Question 10

The ratio of the perimeters of two circles is 5:4. What is the ratio of their radii?

Accepted Answer

Let the radii of the two circles be r_1 and r_2.
Perimeters (circumferences) are C_1 = 2\pi r_1 and C_2 = 2\pi r_2.
Given: \frac{C_1}{C_2} = \frac{5}{4}
=> \frac{2\pi r_1}{2\pi r_2} = \frac{5}{4}
=> \frac{r_1}{r_2} = \frac{5}{4}

So, the ratio of their radii is 5:4. — Since circumference is proportional to radius, the ratio of perimeters equals ratio of radii.

Question 11

Find the area of triangle ADE in Fig. 6.31.

Accepted Answer

To find the area of triangle ADE, use the given figure (Fig. 6.31). Identify the base and height of triangle ADE from the figure. Then apply the formula for the area of a triangle: Area = 1/2 × base × height. Calculate the values accordingly to get the area. — Step 1: Identify base and height of triangle ADE from Fig. 6.31.
Step 2: Use formula Area = 1/2 × base × height.
Step 3: Substitute the values and calculate the area.

Question 12

The parallel sides of a trapezium are 40 cm and 20 cm. If its non-parallel sides are both equal, each being 26 cm, find the area of the trapezium.

Accepted Answer

Given: Parallel sides a = 40 cm, b = 20 cm, non-parallel sides = 26 cm each.

Step 1: Calculate the height (h) of the trapezium using Pythagoras theorem.
Let the trapezium be ABCD with AB and DC parallel, and AD = BC = 26 cm.
Drop perpendiculars from the vertices to find height.

Step 2: Use the formula for area of trapezium:
Area = 1/2 × (sum of parallel sides) × height = 1/2 × (40 + 20) × h = 30 × h.

Step 3: Calculate h from triangle formed by non-parallel sides and difference of parallel sides.

Step 4: Substitute h to find area. — 1. Let the trapezium be ABCD with AB = 40 cm, DC = 20 cm, AD = BC = 26 cm.
2. The difference between parallel sides = 40 - 20 = 20 cm.
3. Drop perpendiculars from B and C to AD to form right triangles.
4. Using Pythagoras theorem, height h = sqrt(26^2 - 10^2) = sqrt(676 - 100) = sqrt(576) = 24 cm.
5. Area = 1/2 × (40 + 20) × 24 = 1/2 × 60 × 24 = 720 cm².

Question 13

Find the area of a triangle, given that its sides are 8 cm and 11 cm long, and its perimeter is 32 cm.

Accepted Answer

Given sides: a = 8 cm, b = 11 cm, perimeter = 32 cm.

Step 1: Find the third side c = perimeter - (a + b) = 32 - (8 + 11) = 13 cm.

Step 2: Use Heron's formula to find area.

Heron's formula: Area = sqrt[s(s - a)(s - b)(s - c)] where s = semi-perimeter = perimeter/2 = 16 cm.

Step 3: Calculate area = sqrt[16(16-8)(16-11)(16-13)] = sqrt[16 × 8 × 5 × 3] = sqrt[1920] ≈ 43.82 cm². — 1. Find third side c = 32 - (8 + 11) = 13 cm.
2. Calculate semi-perimeter s = 32/2 = 16 cm.
3. Apply Heron's formula:
   Area = sqrt[s(s - a)(s - b)(s - c)]
   = sqrt[16 × 8 × 5 × 3]
   = sqrt[1920]
   ≈ 43.82 cm².

Question 14

The sides of a triangular plot are in the ratio 3: 5: 7; its perimeter is 300 m. Find its area.

Accepted Answer

Given ratio of sides = 3:5:7, perimeter = 300 m.

Step 1: Let sides be 3x, 5x, 7x.

Step 2: Sum of sides = 3x + 5x + 7x = 15x = 300 ⇒ x = 20.

Step 3: Sides are 60 m, 100 m, and 140 m.

Step 4: Calculate semi-perimeter s = 300/2 = 150 m.

Step 5: Use Heron's formula:
Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[150(150-60)(150-100)(150-140)] = sqrt[150 × 90 × 50 × 10] = sqrt[6,750,000] ≈ 2598.08 m². — 1. Assign sides as 3x, 5x, 7x.
2. Calculate x = 300/15 = 20.
3. Sides: 60 m, 100 m, 140 m.
4. Semi-perimeter s = 150 m.
5. Apply Heron's formula:
   Area = sqrt[150 × 90 × 50 × 10]
   = sqrt[6,750,000]
   ≈ 2598.08 m².

Question 15

One diagonal of a rhombus is twice as long as the other diagonal. If the rhombus has area 128 cm², find the length of the shorter diagonal.

Accepted Answer

Let the shorter diagonal be d cm. Then the longer diagonal = 2d cm.

Area of rhombus = (1/2) × product of diagonals = 128 cm².

So, (1/2) × d × 2d = 128 ⇒ d² = 128 ⇒ d = sqrt(128) = 8√2 cm.

Therefore, the length of the shorter diagonal is 8√2 cm. — 1. Let shorter diagonal = d.
2. Longer diagonal = 2d.
3. Area = (1/2) × d × 2d = d².
4. Given area = 128 ⇒ d² = 128 ⇒ d = sqrt(128) = 8√2 cm.

Measuring Space: Perimeter and Area

Measuring Space: Perimeter and Area — Study Notes

Introduction

Perimeter of Plane Figures

Area of a Rectangle

Practice Questions — Measuring Space: Perimeter and Area

All 8 Chapters in Mathematics